Group #18011388 — subgroups

Subgroups — #18011388

Order 4736.

348 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558585	C₂	1	4	yes	maximal, complemented
2	#14558583	C₂	1	4	yes	maximal, complemented
3	#14558540	C₂	1	2	yes	maximal
4	#14558537	C₂	1	1	yes	maximal, characteristic
5	#14558498	C₂	1	2	yes	maximal
6	#14558492	C₂	1	1	yes	maximal, characteristic
7	#14558089	C₂	1	1	yes	maximal, characteristic
8	C₂×#10596953	C₂²	1	2	yes	—
9	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
11	#11924603	C₂²	1	4	yes	—
12	C₄⋊#64438	C₂²	1	4	yes	—
13	C₄⋊#64438	C₂²	1	4	yes	—
14	#11924591	C₂²	1	2	yes	—
15	#11924591	C₂²	1	2	yes	—
16	#11924591	C₂²	1	4	yes	—
17	C₄×#64438	C₂²	1	4	yes	—
18	#11924535	C₄	1	2	yes	—
19	#11924528	C₄	1	1	yes	characteristic
20	#11924514	C₄	1	4	yes	—
21	#11924501	C₂²	1	4	yes	—
22	#11924494	C₂²	1	2	yes	—
23	C₄²×D₃₇	C₂²	1	1	yes	characteristic
24	#125×C₃₇	C₄	1	1	yes	characteristic
25	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₇₄	C₂×C₄	1	4	yes	—
28	C₂×#64438	C₂³	1	4	yes	—
29	C₂×#64438	C₂³	1	1	yes	characteristic
30	C₂×#64438	C₂³	1	1	yes	characteristic
31	C₂×#64438	C₂³	1	2	yes	—
32	#10596953	—	2	4	no	—
33	D₃₇⋊C₈	—	2	2	no	—
34	D₃₇⋊C₈	Q₈	1	4	yes	complemented
35	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
36	C₄⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
37	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
38	#10596938	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
40	#10596936	C₂×C₄	1	4	yes	—
41	#10596936	C₂×C₄	1	4	yes	—
42	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₃₇	C₂³	1	2	yes	—
44	C₄×Dic₃₇	C₂³	1	4	yes	—
45	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
50	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
51	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
52	C₄×D₃₇	—	4	2	no	—
53	C₄×D₃₇	C₂×Q₈	1	2	yes	—
54	C₄×D₃₇	#41	1	2	yes	—
55	Dic₇₄	—	4	4	no	—
56	#64438	—	2	1	no	—
57	#64438	—	2	1	no	—
58	#64438	—	2	4	no	—
59	#64438	C₂×Q₈	1	4	yes	—
60	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
61	C₂×D₃₇	—	2	2	no	—
62	C₂×D₃₇	C₄×Q₈	1	2	yes	—
63	C₄×C₃₇	—	2	4	no	—
64	C₄×C₃₇	—	2	2	no	—
65	C₄×C₃₇	C₄×Q₈	1	2	yes	—
66	Dic₃₇	—	2	2	no	—
67	Dic₃₇	—	2	4	no	—
68	Dic₃₇	C₂²×Q₈	1	1	yes	characteristic
69	Dic₃₇	C₂×#41	1	1	yes	characteristic
70	#2961	—	37	1	no	maximal, Sylow
71	C₂×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
72	C₂×C₃₇	#568	1	2	yes	—
73	D₃₇	—	2	2	no	—
74	#533	—	37	1	no	—
75	#446	—	37	4	no	—
76	#446	—	37	4	no	—
77	#424	—	37	2	no	—
78	#423	—	37	1	no	—
79	#405	—	37	2	no	—
80	C₂×#97	—	37	1	no	—
81	C₃₇	#2961	1	1	yes	characteristic, complemented, Sylow
82	C₂×M₁₆	—	37	2	no	—
83	C₂²×C₈	—	37	1	no	—
84	C₂²×C₈	—	37	1	no	—
85	C₄⋊Q₈	—	37	2	no	—
86	#125	—	37	1	no	—
87	#125	#3866	1	1	yes	characteristic
88	C₂²⋊Q₈	—	37	4	no	—
89	C₄×Q₈	—	37	4	no	—
90	#117	—	37	2	no	—
91	C₂×C₄²	—	37	1	no	—
92	C₄⋊C₈	—	37	4	no	—
93	C₄⋊C₈	—	37	4	no	—
94	C₄⋊C₈	—	37	4	no	—
95	#97	—	37	2	no	—
96	#97	—	37	2	no	—
97	#97	—	37	4	no	—
98	C₄×C₈	—	37	4	no	—
99	C₂×Q₈	—	37	4	no	—
100	C₂²×C₄	—	37	1	no	—
101	C₂²×C₄	—	37	2	no	—
102	M₁₆	—	74	4	no	—
103	C₂×C₈	—	37	4	no	—
104	C₂×C₈	—	37	2	no	—
105	C₂×C₈	—	37	4	no	—
106	C₂×C₈	—	37	1	no	—
107	C₂×C₈	—	37	1	no	—
108	C₂×C₈	—	74	2	no	—
109	C₄⋊C₄	—	37	4	no	—
110	C₄⋊C₄	—	37	4	no	—
111	C₄⋊C₄	—	37	2	no	—
112	C₄⋊C₄	C₂×#3866	1	2	yes	—
113	C₄⋊C₄	C₂×#3866	1	4	yes	—
114	C₂²⋊C₄	—	37	4	no	—
115	C₄²	—	37	1	no	—
116	C₄²	—	37	4	no	—
117	C₄²	—	37	2	no	—
118	C₄²	C₂×#3866	1	1	yes	characteristic
119	C₂³	—	37	1	no	—
120	Q₈	—	148	4	no	—
121	C₂×C₄	—	37	1	no	—
122	C₂×C₄	—	37	2	no	—
123	C₂×C₄	—	37	2	no	—
124	C₂×C₄	—	37	2	no	—
125	C₂×C₄	—	37	4	no	—
126	C₂×C₄	—	148	2	no	—
127	C₂×C₄	C₂²×#3866	1	2	yes	—
128	C₂×C₄	C₂²×#3866	1	4	yes	—
129	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
130	C₈	—	37	4	no	—
131	C₈	—	74	1	no	—
132	C₈	—	74	1	no	—
133	C₈	—	74	4	no	—
134	C₂²	—	37	2	no	—
135	C₂²	—	74	2	no	—
136	C₂²	C₂³×#3866	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	37	1	no	—
140	C₄	—	37	1	no	—
141	C₄	—	74	2	no	—
142	C₄	—	74	4	no	—
143	C₄	Q₈×#3866	1	2	yes	—
144	C₂	—	74	2	no	—
145	C₂	C₂×Q₈×#3866	1	1	yes	characteristic
146	C₂	#14558612	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558585

C₂

yes

maximal, complemented

#14558583

C₂

yes

maximal, complemented

#14558540

C₂

yes

maximal

#14558537

C₂

yes

maximal, characteristic

#14558498

C₂

yes

maximal

#14558492

C₂

yes

maximal, characteristic

#14558089

C₂

yes

maximal, characteristic

C₂×#10596953

C₂²

yes

—

C₂×D₃₇⋊C₈

C₂²

yes

characteristic

C₂×D₃₇⋊C₈

C₂²

yes

characteristic

#11924603

C₂²

yes

—

C₄⋊#64438

C₂²

yes

—

C₄⋊#64438

C₂²

yes

—

#11924591

C₂²

yes

—

#11924591

C₂²

yes

—

#11924591

C₂²

yes

—

C₄×#64438

C₂²

yes

—

#11924535

C₄

yes

—

#11924528

C₄

yes

characteristic

#11924514

C₄

yes

—

#11924501

C₂²

yes

—

#11924494

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

#125×C₃₇

C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×#64438

C₂³

yes

—

C₂×#64438

C₂³

yes

characteristic

C₂×#64438

C₂³

yes

characteristic

C₂×#64438

C₂³

yes

—

#10596953

—

D₃₇⋊C₈

—

D₃₇⋊C₈

Q₈

yes

complemented

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

#10596938

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

C₂×C₄

yes

—

#10596936

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂³

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂²×D₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂⁴

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

C₂×Q₈

yes

—

C₄×D₃₇

#41

yes

—

Dic₇₄

—

—

—

—

C₂×Q₈

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₄×Q₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×Q₈

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂²×Q₈

yes

characteristic

Dic₃₇

C₂×#41

yes

characteristic

#2961

—

maximal, Sylow

C₂×C₃₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₃₇

#568

yes

—

D₃₇

—

—

—

—

—

—

—

C₂×#97

—

C₃₇

#2961

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

—

yes

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

C₂×#3866

yes

—

113

C₄⋊C₄

C₂×#3866

yes

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂×#3866

yes

characteristic

119

C₂³

—

120

Q₈

—

148

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

148

—

127

C₂×C₄

C₂²×#3866

yes

—

128

C₂×C₄

C₂²×#3866

yes

—

129

C₂×C₄

C₂²×#3866

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂³×#3866

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

Q₈×#3866

yes

—

144

C₂

—

145

C₂

C₂×Q₈×#3866

yes

characteristic

146

C₂

#14558612

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	#125×C₃₇	C₄
prime radical	C₂²×D₃₇	C₂²×C₄
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	#125	#3866
hypocenter	C₃₇	#2961
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×#3866

commutator, socle

C₂²×C₃₇

C₂³×C₄

Fitting

#125×C₃₇

C₄

prime radical

C₂²×D₃₇

C₂²×C₄

cosocle

C₂×Dic₃₇

C₂⁴

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #14558585	C₂	semidirect
2. #14558583	C₂	semidirect
34. D₃₇⋊C₈	Q₈	semidirect
81. C₃₇	#2961	semidirect

H ≅ G/N

type

1. #14558585

C₂

semidirect

2. #14558583

C₂

semidirect

34. D₃₇⋊C₈

Q₈

semidirect

81. C₃₇

#2961

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14558585	2
#14558583	2
#14558540	2
#14558537	2
#14558498	2
#14558492	2
#14558089	2
#2961	37

subgroup

index

Subgroups — #18011388

Conjugacy classes of subgroups

Complements

Maximal subgroups